Exponential bounds for the minimum contrast with some applications
- Golubev, Yuri
- Spokoiny, Vladimir
2010 Mathematics Subject Classification
- 62F10 62F12 62F25
- risk bound, quasi maximum likelihood, smooth contrast
The paper studies parametric minimum contrast estimates under rather general conditions. The quality if estimation is measured by the rate function related to the contrast which allows for stating the results without specifying the particular parametric structure of the model. This approach permits also to go far beyond the classical i.i.d. case and to obtain nonasymptotic upper bounds for the risk. These bounds apply even for small or moderate samples. They also cover the case of misspecified parametric models. Another important feature of the approach is that it works well in the case when the parametric set can be unbounded and non-compact. In the case of a smooth contrast, the obtained exponential bounds do not rely on the covering numbers and can be easily computed. We also illustrate how these bound can be used for statistical inference: bounding the estimation risk, constructing the confidence sets for the underlying parameters, establishing the concentration properties of the minimum contrast estimate. The general results are specified to the case of a Gaussian contrast and of an i.i.d. sample. We also illustrate the approach by several popular examples including least squares and least absolute deviation contrasts and the problem of estimating the location of the change point. What we obtain in these examples slightly differs from usual asymptotic results known in the classical literature. This difference is due to the unboundness of the parameter set and a possible model misspecification.
- Electron. J. Stat., 3 (2009) pp. 712--746.